Some textbooks on algebraic topology define a simplicial complex as a special kind of subset of a Euclidean space (e.g., A. Wallace, p.3). Others define an abstract simplicial complex as a special kind of combinatorial structure (e.g., E. H. Spanier, p. 108). There exists a functor from the category of abstract simplicial complexes and simplicial maps to the category of topological spaces and continuous functions. My question is, what is an *algebraic* criterion for deciding whether or not there exists an embedding of the realization of an abstract simplicial complex into a Euclidean space of *specified* dimension. For example (J. R. Munkres, p. 18) exhibits an abstract simplicial complex whose realization is a Klein bottle, which cannot be embedded in $R^3$.

If a concrete 2-dimensional finite simplicial complex consists of point, line segment, and triangular subsets of 3-dimensional Euclidean space, then obviously there is more information about these simplices and their relationships than is contained in the corresponding abstract simplicial complex. In particular, there are the lengths of the line segments, the angles between connected line segments (one shared point), and the dihedral angles between connected triangles (one shared edge). This is a finite collection of data. My actual motivation for the titular question is the following question. Given an abstract finite simplicial complex together with "lengths" of 1-dimensional abstract simplices, "angles" between connected 1-dimensional abstract simplices, and "dihedral angles" between connected 2-dimensional abstract simplices, is there an algorithm to decide whether there is an embedding of its simplicial realization in 3-dimensional Euclidean space?

manifoldsinto Euclidean space? Abstract simplicial complexes should be thought of as a very broad class of spaces from an algebraic topology perspective. For finite complexes, you can get crude bounds based on the number of simplices. But for anything more refined, I think you'll have to restrict attention to more specific classes of simplicial complexes. $\endgroup$